Subgroups — C₂²×D₅×#95 #12257466

Order 1280.

14334 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₄×D₅	C₂	1	3	yes	maximal
2	C₂²×C₅×#95	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#3963	C₂	1	3	yes	maximal, complemented
4	C₂×D₅×#95	C₂	1	24	yes	maximal, direct product factor
5	C₂⁵×D₅	C₂²	1	1	yes	characteristic
6	C₂⁴×C₄×C₅	C₂²	1	3	yes	—
7	C₂⁴×Dic₅	C₂²	1	3	yes	—
8	C₂³×C₄×D₅	—	2	48	no	—
9	C₂³×C₄×D₅	C₄	1	48	yes	complemented
10	C₂³×C₄×D₅	C₂²	1	36	yes	—
11	C₂×C₅×#95	C₂²	1	12	yes	complemented
12	C₂×#3963	C₂²	1	36	yes	complemented
13	D₅×#95	C₂²	1	64	yes	direct product factor
14	C₂³×#95	—	5	1	no	maximal, Sylow
15	C₂⁵×C₅	C₂³	1	1	yes	characteristic
16	C₂⁴×D₅	C₂×C₄	1	24	yes	—
17	C₂⁴×D₅	D₄	1	24	yes	—
18	C₂⁴×D₅	Q₈	1	8	yes	—
19	C₂⁴×D₅	C₂³	1	6	yes	—
20	C₂³×C₄×C₅	—	2	24	no	—
21	C₂³×C₄×C₅	C₂×C₄	1	24	yes	complemented
22	C₂³×C₄×C₅	C₂³	1	18	yes	—
23	C₂³×Dic₅	—	2	24	no	—
24	C₂³×Dic₅	C₂×C₄	1	24	yes	complemented
25	C₂³×Dic₅	C₂³	1	18	yes	—
26	C₂²×C₄×D₅	—	2	384	no	—
27	C₂²×C₄×D₅	—	2	288	no	—
28	C₂²×C₄×D₅	C₂×C₄	1	288	yes	complemented
29	C₂²×C₄×D₅	C₂³	1	48	yes	—
30	C₅×#95	C₂³	1	16	yes	complemented
31	#3963	C₂³	1	48	yes	complemented
32	C₂⁵×C₄	—	5	3	no	—
33	C₂²×#95	—	5	24	no	—
34	C₂²×#95	—	5	3	no	—
35	C₂²×#95	D₅	1	1	yes	characteristic, direct product factor
36	C₂⁴×C₅	C₂²×C₄	1	12	yes	—
37	C₂⁴×C₅	C₂×D₄	1	12	yes	—
38	C₂⁴×C₅	C₂×Q₈	1	4	yes	—
39	C₂⁴×C₅	C₂⁴	1	3	yes	—
40	C₂³×D₅	C₄²	1	64	yes	—
41	C₂³×D₅	C₂²⋊C₄	1	192	yes	—
42	C₂³×D₅	C₄⋊C₄	1	192	yes	—
43	C₂³×D₅	C₂²×C₄	1	72	yes	—
44	C₂³×D₅	C₂×D₄	1	72	yes	—
45	C₂³×D₅	C₂×Q₈	1	24	yes	—
46	C₂³×D₅	C₂⁴	1	4	yes	—
47	C₂²×C₄×C₅	—	2	96	no	—
48	C₂²×C₄×C₅	—	2	72	no	—
49	C₂²×C₄×C₅	C₂²×C₄	1	72	yes	complemented
50	C₂²×C₄×C₅	C₂⁴	1	12	yes	—
51	C₂²×Dic₅	—	2	96	no	—
52	C₂²×Dic₅	—	2	72	no	—
53	C₂²×Dic₅	C₂²×C₄	1	72	yes	complemented
54	C₂²×Dic₅	C₂⁴	1	12	yes	—
55	C₂×C₄×D₅	—	2	1152	no	—
56	C₂×C₄×D₅	—	2	192	no	—
57	C₂×C₄×D₅	C₂²×C₄	1	192	yes	complemented
58	C₂⁶	—	5	1	no	—
59	C₂⁴×C₄	—	5	3	no	—
60	C₂⁴×C₄	—	5	48	no	—
61	C₂⁴×C₄	—	5	36	no	—
62	C₂⁴×C₄	—	10	48	no	—
63	C₂⁴×C₄	C₂×D₅	1	3	yes	—
64	C₂×#95	—	5	36	no	—
65	C₂×#95	—	5	64	no	—
66	C₂×#95	C₂×D₅	1	12	yes	direct product factor
67	C₂³×C₅	C₂×C₄²	1	16	yes	—
68	C₂³×C₅	C₂×C₂²⋊C₄	1	48	yes	—
69	C₂³×C₅	C₂×C₄⋊C₄	1	48	yes	—
70	C₂³×C₅	C₂³×C₄	1	18	yes	—
71	C₂³×C₅	C₂²×D₄	1	18	yes	—
72	C₂³×C₅	C₂²×Q₈	1	6	yes	—
73	C₂³×C₅	C₂⁵	1	1	yes	characteristic
74	C₂²×D₅	#95	1	512	yes	direct product factor
75	C₂²×D₅	C₂×C₄²	1	96	yes	—
76	C₂²×D₅	C₂×C₂²⋊C₄	1	288	yes	—
77	C₂²×D₅	C₂×C₄⋊C₄	1	288	yes	—
78	C₂²×D₅	C₂³×C₄	1	24	yes	—
79	C₂²×D₅	C₂²×D₄	1	24	yes	—
80	C₂²×D₅	C₂²×Q₈	1	8	yes	—
81	C₂×C₄×C₅	—	2	144	no	—
82	C₂×C₄×C₅	—	2	24	no	—
83	C₂×C₄×C₅	C₂³×C₄	1	24	yes	complemented
84	C₂×Dic₅	—	2	144	no	—
85	C₂×Dic₅	—	2	24	no	—
86	C₂×Dic₅	C₂³×C₄	1	24	yes	complemented
87	C₄×D₅	—	2	384	no	—
88	C₂⁵	—	5	24	no	—
89	C₂⁵	—	5	24	no	—
90	C₂⁵	—	5	8	no	—
91	C₂⁵	—	5	6	no	—
92	C₂⁵	C₂²×D₅	1	1	yes	characteristic
93	C₂³×C₄	—	2	24	no	—
94	C₂³×C₄	—	5	24	no	—
95	C₂³×C₄	—	5	18	no	—
96	C₂³×C₄	—	5	288	no	—
97	C₂³×C₄	—	5	48	no	—
98	C₂³×C₄	—	10	288	no	—
99	C₂³×C₄	—	10	384	no	—
100	C₂³×C₄	—	10	24	no	—
101	C₂³×C₄	C₄×D₅	1	24	yes	complemented
102	C₂³×C₄	C₂²×D₅	1	18	yes	—
103	#95	—	5	48	no	—
104	#95	C₂²×D₅	1	16	yes	direct product factor
105	C₂²×C₅	C₂×#95	1	64	yes	complemented
106	C₂²×C₅	C₂²×C₄²	1	12	yes	—
107	C₂²×C₅	C₂²×C₂²⋊C₄	1	36	yes	—
108	C₂²×C₅	C₂²×C₄⋊C₄	1	36	yes	—
109	C₂²×C₅	C₂⁴×C₄	1	3	yes	—
110	C₂²×C₅	C₂³×D₄	1	3	yes	—
111	C₂²×C₅	C₂³×Q₈	1	1	yes	characteristic
112	C₂×D₅	C₂×#95	1	384	yes	direct product factor
113	C₂×D₅	C₂²×C₄²	1	16	yes	—
114	C₂×D₅	C₂²×C₂²⋊C₄	1	48	yes	—
115	C₂×D₅	C₂²×C₄⋊C₄	1	48	yes	—
116	C₄×C₅	—	2	24	no	—
117	Dic₅	—	2	24	no	—
118	C₂⁴	—	5	72	no	—
119	C₂⁴	—	5	72	no	—
120	C₂⁴	—	5	24	no	—
121	C₂⁴	—	5	4	no	—
122	C₂⁴	—	5	64	no	—
123	C₂⁴	—	5	192	no	—
124	C₂⁴	—	5	192	no	—
125	C₂⁴	C₂×C₄×D₅	1	12	yes	—
126	C₂⁴	D₄×D₅	1	12	yes	—
127	C₂⁴	Q₈×D₅	1	4	yes	—
128	C₂⁴	C₂³×D₅	1	3	yes	—
129	C₂²×C₄	—	2	72	no	—
130	C₂²×C₄	—	2	96	no	—
131	C₂²×C₄	—	5	72	no	—
132	C₂²×C₄	—	5	12	no	—
133	C₂²×C₄	—	5	192	no	—
134	C₂²×C₄	—	10	1152	no	—
135	C₂²×C₄	—	10	192	no	—
136	C₂²×C₄	—	10	72	no	—
137	C₂²×C₄	—	10	96	no	—
138	C₂²×C₄	C₂×C₄×D₅	1	72	yes	complemented
139	C₂²×C₄	C₂³×D₅	1	12	yes	—
140	C₂×C₅	C₂²×#95	1	24	yes	complemented
141	C₂×C₅	C₂³×C₄²	1	1	yes	characteristic
142	C₂×C₅	C₂³×C₂²⋊C₄	1	3	yes	—
143	C₂×C₅	C₂³×C₄⋊C₄	1	3	yes	—
144	D₅	C₂²×#95	1	32	yes	direct product factor
145	C₂³	—	5	96	no	—
146	C₂³	—	5	288	no	—
147	C₂³	—	5	288	no	—
148	C₂³	—	5	512	no	—
149	C₂³	—	5	24	no	—
150	C₂³	—	5	24	no	—
151	C₂³	—	5	8	no	—
152	C₂³	C₄²×D₅	1	16	yes	—
153	C₂³	D₅×C₂²⋊C₄	1	48	yes	—
154	C₂³	D₅×C₄⋊C₄	1	48	yes	—
155	C₂³	C₂²×C₄×D₅	1	18	yes	—
156	C₂³	C₂×D₄×D₅	1	18	yes	—
157	C₂³	C₂×Q₈×D₅	1	6	yes	—
158	C₂³	C₂⁴×D₅	1	1	yes	characteristic
159	C₂×C₄	—	2	144	no	—
160	C₂×C₄	—	2	24	no	—
161	C₂×C₄	—	5	24	no	—
162	C₂×C₄	—	10	384	no	—
163	C₂×C₄	—	10	144	no	—
164	C₂×C₄	—	10	24	no	—
165	C₂×C₄	C₂²×C₄×D₅	1	24	yes	complemented
166	C₅	C₂³×#95	1	1	yes	characteristic, complemented, Sylow
167	C₂²	—	5	48	no	—
168	C₂²	—	5	384	no	—
169	C₂²	—	5	16	no	—
170	C₂²	—	5	48	no	—
171	C₂²	D₅×#95	1	64	yes	direct product factor
172	C₂²	C₂×C₄²×D₅	1	12	yes	—
173	C₂²	C₂×D₅×C₂²⋊C₄	1	36	yes	—
174	C₂²	C₂×D₅×C₄⋊C₄	1	36	yes	—
175	C₂²	C₂³×C₄×D₅	1	3	yes	—
176	C₂²	C₂²×D₄×D₅	1	3	yes	—
177	C₂²	C₂²×Q₈×D₅	1	1	yes	characteristic
178	C₄	—	2	24	no	—
179	C₄	—	10	24	no	—
180	C₂	—	5	32	no	—
181	C₂	C₂×D₅×#95	1	24	yes	direct product factor
182	C₂	C₂²×C₄²×D₅	1	1	yes	characteristic
183	C₂	C₂²×D₅×C₂²⋊C₄	1	3	yes	—
184	C₂	C₂²×D₅×C₄⋊C₄	1	3	yes	—

	subgroup	G/H
center	C₂⁵	C₂²×D₅
commutator	C₂×C₅	C₂³×C₄²
Frattini	C₂³	C₂⁴×D₅
Fitting	C₂²×C₅×#95	C₂
socle	C₂⁵×C₅	C₂³
prime radical	C₂⁵×D₅	C₂²
cosocle	C₂³×C₅	C₂⁵
hypercenter	C₂²×#95	D₅
hypocenter	C₅	C₂³×#95
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂⁵ ◁ C₂²×#95

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂²×C₅×#95	C₂	semidirect
3. C₂²×#3963	C₂	semidirect
4. C₂×D₅×#95	C₂	direct (N × H)
9. C₂³×C₄×D₅	C₄	semidirect
11. C₂×C₅×#95	C₂²	semidirect
12. C₂×#3963	C₂²	semidirect
13. D₅×#95	C₂²	direct (N × H)
21. C₂³×C₄×C₅	C₂×C₄	semidirect
24. C₂³×Dic₅	C₂×C₄	semidirect
28. C₂²×C₄×D₅	C₂×C₄	semidirect
30. C₅×#95	C₂³	semidirect
31. #3963	C₂³	semidirect
35. C₂²×#95	D₅	direct (N × H)
49. C₂²×C₄×C₅	C₂²×C₄	semidirect
53. C₂²×Dic₅	C₂²×C₄	semidirect
57. C₂×C₄×D₅	C₂²×C₄	semidirect
66. C₂×#95	C₂×D₅	direct (N × H)
74. C₂²×D₅	#95	direct (N × H)
83. C₂×C₄×C₅	C₂³×C₄	semidirect
86. C₂×Dic₅	C₂³×C₄	semidirect
101. C₂³×C₄	C₄×D₅	semidirect
104. #95	C₂²×D₅	direct (N × H)
105. C₂²×C₅	C₂×#95	semidirect
112. C₂×D₅	C₂×#95	direct (N × H)
138. C₂²×C₄	C₂×C₄×D₅	semidirect
140. C₂×C₅	C₂²×#95	semidirect
144. D₅	C₂²×#95	direct (N × H)
165. C₂×C₄	C₂²×C₄×D₅	semidirect
166. C₅	C₂³×#95	semidirect
171. C₂²	D₅×#95	direct (N × H)
181. C₂	C₂×D₅×#95	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂⁴×C₄×D₅	2
C₂²×C₅×#95	2
C₂²×#3963	2
C₂×D₅×#95	2
C₂³×#95	5

Subgroups — C22×D5×#95 #12257466

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×D₅×#95 #12257466